The axially symmetric flow of a viscous fluid between two infinite rotating disks

1962 ◽  
Vol 266 (1324) ◽  
pp. 109-121 ◽  
Keyword(s):  
Stokes Equations ◽  
Transverse Velocity ◽  
Complete Solution ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Rotating Disks ◽  
Symmetric Flow ◽  
Navier Stokes Equations ◽  
High Reynolds

This is a numerical investigation of the similarity solutions of the Navier-Stokes equations describing the steady axially symmetric flow of a viscous incompressible fluid between two infinite rotating disks. Several cases have been examined in detail and the radial and transverse velocity profiles are displayed; value of the torque experienced in these cases are also given. It is found that at high Reynolds numbers, the main core of the fluid is in a state of solid rotation for practically all values of the ratio of angular velocity of the two disks. When the disks are rotating in the same sense, and when one is at rest and the other is rotating, the results show that edge effects must be taken into account in any complete solution to the problem. However, when the disks rotate in opposite directions, the solutions exhibit features which appear unlikely to occur in practice.

High Reynolds number flow between two inflnite rotating disks

1971 ◽  
Vol 12 (4) ◽  
pp. 483-501 ◽  
Author(s):  
H. Rasmussen
Keyword(s):  
Stokes Equations ◽  
Axisymmetric Flow ◽  
Radial Distance ◽  
Rotating Disk ◽  
Navier Stokes ◽  
Rotating Disks ◽  
Axial Velocity Component ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
High Reynolds

In 1921 von Karman [1] showed that the Navier-Stokes equations for steady viscous axisymmetric flow can be reduced to a set of ordinary differential equations if it is assumed that the axial velocity component is independent of the radial distance from the axis of symmetry. He used these similarity equations to obtain a solution for the flow near an infinite rotating disk. Later Batchelor [2] and Stewartson [3] applied these equations to the problem of steady flow between two infinite disks rotating in parallel planes a finite distance apart.

A nonexistence result for axially symmetric flows with constant angular velocities at infinity

1983 ◽  
Vol 93 (3-4) ◽  
pp. 229-231
Author(s):  
Alan R. Elcrat ◽  
David Siegel
Keyword(s):  
Boundary Value Problem ◽  
Stokes Equations ◽  
Rotating Disk ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Rotating Disks ◽  
Navier Stokes Equations ◽  
Nonexistence Result ◽  
Angular Velocities ◽  
Made In

SynopsisIf von Kármán's substitution is made in the Navier-Stokes equations, and boundary conditions corresponding to a flow in all of space with constant angular velocities at infinity are imposed, a boundary value problem analgous to those for flow above a rotating disk and between rotating disks is obtained. It is shown here that this problem has no solution.

scholarly journals Transitional vortices in wide-gap spherical annulus flow

2005 ◽  
Vol 2005 (18) ◽  
pp. 2913-2932 ◽  
Author(s):  
E. O. Ifidon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Outer Sphere ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Navier Stokes Equations ◽  
Asymmetric Vortex ◽  
Concentric Spheres ◽  
Wide Gap ◽  
High Reynolds

We develop a semianalytic formulation suitable for solving the Navier-Stokes equations governing the induced steady, axially symmetric motion of an incompressible viscous fluid confined in a wide gap between two differentially rotating concentric spheres. The method is valid for arbitrarily high Reynolds number and aids in the presentation of multiple steady-state flow patterns and their bifurcations. In the case of a rotating inner sphere and a stationary outer sphere, linear stability analysis is conducted to determine whether or not the computed solutions are stable. It is found that the solution transforms smoothly into an unstable solution beginning with asymmetric vortex pairs identified near the point of a symmetry-breaking bifurcation which occurs at Reynolds number 589. This solution transforms smoothly into an unstable asymmetric vortex solution as the Reynolds number increases. Flow modes whose branches have not been previously reported are found using this method. The origin of the flow modes obtained are discussed using bifurcation theory.

Free-stream coherent structures in parallel boundary-layer flows

2014 ◽  
Vol 752 ◽  
pp. 602-625 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Coherent Structures ◽  
Free Stream ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Homotopy Continuation ◽  
Boundary Layer Flows ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractOur concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier–Stokes equations in a layer located a distance of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(\ln \mathit{Re})$ from the wall. Here $\mathit{Re}$ is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

The Onset of Turbulence: Insights Into the Navier-Stokes Equations

2017 ◽  
Author(s):  
Carl E. Rathmann
Keyword(s):  
Stokes Equations ◽  
Direct Consequence ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Fluid Behavior ◽  
Onset Of Turbulence ◽  
And Mathematics ◽  
High Reynolds

For well over 150 years now, theoreticians and practitioners have been developing and teaching students easily visualized models of fluid behavior that distinguish between the laminar and turbulent fluid regimes. Because of an emphasis on applications, perhaps insufficient attention has been paid to actually understanding the mechanisms by which fluids transition between these regimes. Summarized in this paper is the product of four decades of research into the sources of these mechanisms, at least one of which is a direct consequence of the non-linear terms of the Navier-Stokes equation. A scheme utilizing chaotic dynamic effects that become dominant only for sufficiently high Reynolds numbers is explored. This paper is designed to be of interest to faculty in the engineering, chemistry, physics, biology and mathematics disciplines as well as to practitioners in these and related applications.

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